Dual-Intuitionistic Logic

Abstract

The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To obtain a more precise mapping, both LJ and LDJ are extended by adding a "pseudo-difference" operator ∸ which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems ${\bf LJ}^{∸}$ and ${\bf LDJ}^{∸}$, and a simply consistent but $\omega$-inconsistent Set Theory with Unrestricted Comprehension Schema based on LDJ is sketched.